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Problem 802

AIME late

Combinatorics Difficulty 5.5 Find the answer

# 6. CONDITION

In a row from left to right, 13 weights with masses of 1 g, 2 g, 3 g, ..., 13 g were placed. Only seven of them, standing in a row (their order has not changed), remain, while the other six weights have been lost. Can the masses of the remaining weights be determined in two weighings on a balance scale? Justify your answer. It is impossible to distinguish the weights in any way other than by weighing them on the scale.

Your answer

Official solution

Solution. Let the weight of the leftmost of the remaining weights be x. It is sufficient to determine what x is. In the first weighing, we compare the three leftmost weights and the two immediately following them. If the weights are equal, then we have $x+x+1+x+2$ $=\mathrm{x}+3+\mathrm{x}+4$ , i.e., $\mathrm{x}=4$ and everything is established. Suppose the three weights outweigh. Then $\mathrm{x} < 4$ . In the second weighing, we compare the weight of the three leftmost and the weight of the rightmost and the middle weight ( $3 x+3$ and $2 x+9$ ). Equality of weights indicates that $x=6$ , if the pan with the three weights outweighs, then $x=7$ , otherwise $x=5$ .

Answer: it is possible.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.